Let n be a positive integer. If `1/n` has a terminating decimal expansion, then which one of the following is true?

Options

A. n is of the form `5^(x)` , where x is a positive integer
B. n is of the form `2^y`, where y is non-positive x and y
C. n is of the form `2^x . 5^y` for some non-negative x and y
D. n is of the form `10^z` for some positive integer z

Answer

To determine the conditions under which \( \frac{1}{n} \) has a terminating decimal expansion, we need to analyze the properties of the integer \( n \).

### Step-by-Step Solution:

1. **Understanding Terminating Decimals**:
A fraction \( \frac{a}{b} \) has a terminating decimal expansion if and only if the denominator \( b \) (in its simplest form) has no prime factors other than 2 and 5. This means that \( b \) can be expressed in the form \( 2^x \cdot 5^y \) where \( x \) and \( y \) are non-negative integers.

2. **Identifying \( n \)**:
Since we are considering \( \frac{1}{n} \), we need to ensure that \( n \) meets the criteria for having only the prime factors 2 and 5.

3. **Form of \( n \)**:
Therefore, \( n \) must be expressible in the form:
\[ n = 2^x \cdot 5^y \]
where \( x \) and \( y \) are non-negative integers (including zero).

4. **Conclusion**:
Hence, for \( \frac{1}{n} \) to have a terminating decimal expansion, \( n \) must be of the form \( 2^x \cdot 5^y \).

5. **Checking Options**:
If the options provided include a statement like "n is of the form \( 2^x \cdot 5^y \) for some non-negative integers \( x \) and \( y \)", then that option is correct.

### Final Answer:
The correct statement is that \( n \) must be of the form \( 2^x \cdot 5^y \) for some non-negative integers \( x \) and \( y \).